A370206 -id:A370206 - cp's OEIS frontend

A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

78, 820, 6328, 62128, 5539456, 155155972096, 739936, 69342976, 431056, 31494016, 44864128, 3525354496, 3788128

Offset: 2

Hartmut F. W. Hoft, Feb 11 2024

nonn

a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.

Conjecture: The sequence is infinite.

			a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that  2^k < p < 2^(k+1) need not be true.
a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
a(16) = 24129129742336 = 2^16 * 53 * 6946817.
Table of records of number of digits a(2) through a(500):
sequence index    2  3  4  5  6   7   15    76   419    438
number of digits  2  3  4  5  7  12  355  3854  5856  20049

Chai Wah Wu, Table of n, a(n) for n = 2..75 [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - _N. J. A. Sloane_, Feb 18 2024]

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205, A370206.

minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/;PrimeQ[p]
a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/;PrimeQ[p]
Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)

from itertools import count
from sympy import prime, isprime
def A370209(n):
    p = prime(n)
    return next((p<Chai Wah Wu, Feb 17 2024

a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.

A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

Original entry on oeis.org

6, 12, 78, 20, 102, 1014, 24, 114, 1734, 12246, 28, 138, 2166, 12714, 171366, 40, 174, 3174, 13026, 501126, 1922622, 48, 186, 5046, 13182, 781926, 2057406, 28960854, 56, 222, 5766, 13494, 1679046, 2067546, 144825414, 300014754, 80, 246, 8214, 13962, 4243686, 2072382, 282275286, 300137214, 4174476774

Offset: 1

Hartmut F. W. Hoft, Apr 21 2024

Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
d_(2i+2)  < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
with the odd divisors d_j of n in increasing order.

			a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
  -------------------------------------------------------------------------
   n\m  1    2     3     4       5         6          7          8
  -------------------------------------------------------------------------
   1|   6   78   1014  12246   171366   1922622    28960854  300014754 ...
   2|  12  102   1734  12714   501126   2057406   144825414  300137214 ...
   3|  20  114   2166  13026   781926   2067546   282275286  300235182 ...
   4|  24  138   3174  13182   1679046  2072382   888215334  300357642 ...
   5|  28  174   5046  13494   4243686  2081742  3568939926  300431118 ...
   6|  40  186   5766  13962   5541126  2091882     ...      300602562 ...
   7|  48  222   8214  14118   8487372  2097966              300651546 ...
   8|  56  246  10086  14898  11082252  2110134              300896466 ...
   9|  80  258  10092  15054  11244966  2112162              301165878 ...
  10|  88  282  11094  15366  16954566  2116218              301386306 ...
  ...

Row 1 gives A372181.
Column 1 gives A370205.
Column 2 gives A370206.
Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.

divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2

T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.

T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200

Offset: 1

Hartmut F. W. Hoft, Aug 21 2024

nonn

Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

			a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.
a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.
Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).
This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.
A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.
Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.
For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.
Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.

			a(5) = 10 is in the sequence since SRS(10) = {9, 9} consists of 2 parts of width 1 and of sizes 9 = (2^2 - 1)(5+1)/2.
a(15) = 25 is in the sequence since the first part of SRS(25) = {13, 5, 13} has width 1 and has size 13 = (2^1 - 1)(25+1)/2.
a(28) = 44 is in the sequence since SRS(44) = {42, 42} has width 1 and has size 42 = (2^3 - 1)(11+1)/2.
The upper left hand 11 X 11 section of array T(j, k) shows the j-th number m in this sequence of the form m = 2^k * q with q odd. The first part of SRS(m) of every number in column k consists of 2^(k+1) - 1 legs of width 1.
j\k| 0   1   2    3    4     5     6      7      8       9       10  ...
------------------------------------------------------------------------
1  | 3   10  44   136  592   2144  8384   32896  133376  527872  2102272
2  | 5   14  52   152  656   2272  8768   33664  133888  528896  2112512
3  | 7   22  68   184  688   2336  8896   34432  138496  531968  2118656
4  | 9   26  76   232  752   2528  9536   34688  140032  537088  2130944
5  | 11  34  92   248  848   2656  9664   35456  142592  538112  2132992
6  | 13  38  116  296  944   2848  10048  35968  144128  543232  2137088
7  | 15  46  124  328  976   3104  10432  36224  145664  544256  2139136
8  | 17  50  148  344  1072  3232  10688  37504  146176  547328  2149376
9  | 19  58  164  376  1136  3296  11072  39296  147712  556544  2161664
10 | 21  62  172  424  1168  3424  11456  39808  150272  558592  2163712
11 | 23  70  188  472  1264  3488  11584  40064  151808  559616  2180096
...
Row 1 is A246956(n), n>=1.
Column 0 is A005408(n) with T(j + 1, 0) = T(j, 0) + 2, n>=1.
Column 1 is A091999(n) with T(j + 2, 1) = T(j, 1) + 12, n>=2.
Column 2 is A270298(n) with T(j + 48, 2) = T(j, 2) + 840, n>=1.
Column 3 is A270301(n) with T(j + 5760, 3) = T(j, 3) + 240240, n>=1.

Cf. A000079, A005408, A091999, A235791, A237287, A237591, A237593, A238524, A246956, A270298, A270301, A341969, A370206, A377654.

(* partsSRS[] and widthPattern[ ] are defined in A377654 *)
a378471[m_, n_] := Select[Range[m, n], Length[partsSRS[#]]>1&&widthPattern[#][[1;;2]]=={1, 0}&]
a378471[1, 105]

A376333 Numbers m whose symmetric representation of sigma(m), SRS(m), consists of widths 0, 1, and 2.

Original entry on oeis.org

15, 35, 45, 63, 70, 75, 77, 78, 91, 99, 102, 105, 110, 114, 117, 130, 135, 138, 143, 153, 154, 165, 170, 174, 175, 182, 186, 187, 189, 190, 195, 209, 221, 222, 225, 231, 238, 245, 246, 247, 255, 258, 266, 273, 282, 285, 286, 297, 299, 318, 322, 323, 325, 345, 348, 350

Offset: 1

Hartmut F. W. Hoft, Sep 20 2024

nonn

Sequence a(n) is the subsequence of A375611 for which the symmetric representation of sigma(a(n)) has at least two parts. The width at the diagonal can be any of the 3 widths.

Let m = 2^k * q, k >= 0 and q odd, be a number in this sequence. Let c be the number of divisors s <= A003056(m) of q for which there is at most one divisor t of q satisfying s < t <= min( 2^(k+1) * s, A003056(m). Let w be the number of times width 2 occurs in the width pattern of m (row m in the triangle of A341960). Then c = (w + 1)/2 when the width at the diagonal is equal to 2 and c = w/2 otherwise.

			SRS(a(1)) consists of 3 parts, its width pattern is 1 0 1 2 1 0 1, and c = 1 with divisor 3.
a(6) = 75 is the smallest number in this sequence which has width  0 on the diagonal; SRS(75) has 4 parts.
a(8) = 78 is the smallest number in this sequence with width pattern 1 2 1 0 1 2 1 (see A370206 and A370209).
a(35) = 225 is the smallest number in the sequence with width 1 on the diagonal; its width pattern is 1 0 1 2 1 2 1 2 1 2 1 2 1 2 1 0 1; w = 6 and c = 3 with divisors 3, 5, and 9.

Cf. A237593, A249223, A341969, A370206, A370209, A375611.

(* function sDiv[ ] is defined in A375611 *)
m012Q[n_] := Union[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]=={0, 1, 2}
a376333[m_, n_] := Select[Range[m, n], m012Q]
a376333[1, 350]

A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

Original entry on oeis.org

3, 78, 10728, 28920, 53752896, 4157280, 278628512256, 90323520, 1658908800, 21499810560, 7487812494923563008, 13005699840, 155267279705546496147456, 111451576596480, 8599694054400, 468208581120, 4172630516011611848266349543424, 5202323481600, 21630916595004029113587563614961664, 67421367982080

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Let the nonincreasing multiset cL = { c_1, ... , c_s } be a factorization of n, let dL = { d_1, ... , d_s } be any set of s distinct odd primes, let q = dL^(cL - 1) = d_1^(c_1 - 1) * ... * d_s^(c_s - 1), and let k satisfy 2^k < q < 2^(k+1). Then SRS(2^k * q) is unimodal of maximum height n, 2^k * q has 2n odd divisors and its width pattern has 2n-1 entries. The smallest possible choice for 2^k * q is with the increasing sequence of odd primes d_i = p_(i+1), 1 <= i <= s. The overall smallest 2^k * q is the minimum among all factorizations of n. The smallest number m for which SRS(m) has two unimodal parts of maximum width n requires the additional prime factor r > 2^(k+1) * q which yields m = 2^k * q * r.

This sequence is column 2 in the array of A367377 and a(2) = A370206(1).

			a(2) = 78  is in the sequence since SRS(78) consists of two parts with width pattern 1 2 1 0 1 2 1 and 78 is the smallest number with those properties.
a(3) = 10728 = 2^3 * 3^2 * 149 is in the sequence since SRS(10728) consists of two parts with width pattern 1 2 3 2 1 0 1 2 3 2 1 and 10728 is the smallest number with those properties.
a(6) = 4157280 = 2^5 * 3^2 * 5 * 2887 is in the sequence. The two factorizations of 6 are {6} and {3, 2} so that with 3^5 = 243 and 3^2 * 5^1 = 45 the inequality 2^5 < 45 < 2^6 determines the single unimodular SRS(32 * 45) of maximum width 6, A250071(6) = 1440. Since 2887 is the smallest prime exceeding 2^6 * 3^2 * 5, 4157280 is the smallest number with SRS(4157280) consisting of two unimodular parts of maximum width 6.

Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.

(* function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
a378470[n_] := Map[f2U, Range[n]]
a378470[20]

a(p) = 2^k * 3^(p-1) * r, for odd primes p, with 2^k < 3^(p-1) < 2^(k+1) and r > 2^(k+1) * 3^(p-1) least prime, i.e., k = floor( (p-1)*(log_2 (3)) ) and r = prime( primepi(2^(k+1) * 3^(p-1)) + 1 ).

cp's OEIS Frontend

A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

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A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

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A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

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A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

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A376333 Numbers m whose symmetric representation of sigma(m), SRS(m), consists of widths 0, 1, and 2.

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A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

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